It is well known that crystal plasticity is the result of the presence and motion of dislocations in a body, manifesting itself through stress, permanent and recoverable deformation, and work hardening of the material. Stress and recoverable deformation arise due to the elastic lattice stretching induced by applied loads and the presence of dislocations; permanent deformation is the result of the motion of dislocations through the lattice; and hardening arises from short and long range elastic interactions, i.e. combine stress effects of a distribution of dislocations in the body, that tend to decrease the ductility (increased strength) of the material.
Despite the many successes of conventional and gradient theories of single crystal plasticity, a field theory that allows for the calculation of stress fields of evolutionary 3-d dislocation distributions in finite bodies under nominally unrestricted deformations and loads, large lattice rotations and nonlinear crystal elasticity is still a research goal.
This presentation elaborates on the development of such a theory and presents some of the simplest consequences of it that seem to suggest that the theory has the potential of describing dislocation plasticity at small scales. In particular, stress fields of a screw dislocation in a linear elastic and neo-Hookean material, the development of a slip step due to exit of an edge dislocation from a crystal, kinematics of rectilinear motion of a straight dislocation and expansion of a polygonal loop, and the kinematics of the initiation of bowing and cross-slip of dislocation segments will be discussed. Driving forces for dislocation motion representing the nature of Schmid and non-Schmid effects arising from dislocation behavior as well as a non-local driving force for dislocation nucleation, as these arise as (non-local) thermodynamic consequences of the theory, will also be discussed.
But what of observed plasticity at coarser scales of resolution? Is it possible to invoke some homogenization procedure for nonlinear evolutionary field equations that may be used to derive a coarse scale theory corresponding to the fine scale dislocation mechanics described above? Time permitting, a candidate procedure based on an application of Muncaster's 'Formal theory of invariant manifolds in mechanics' will be illustrated through two simple examples from phenomenological plasticity and dislocation mechanics.